Central limit theorems for discretized occupation time functionals

TL;DR

This paper establishes central limit theorems for discretized occupation time functionals of Itô semimartingales, demonstrating the efficiency of the trapezoidal rule in achieving minimal asymptotic variance.

Contribution

It introduces novel Fourier domain approximations and proves CLTs for fractional smoothness functions, highlighting the efficiency of specific quadrature rules.

Findings

Trapezoidal rule is rate optimal and efficient.

Explicit $L^2$-lower bound for approximation error.

Central limit theorems established for fractional smoothness functions.

Abstract

The approximation of integral type functionals is studied for discrete observations of a continuous It\^o semimartingale. Based on novel approximations in the Fourier domain, central limit theorems are proved for $L^2$-Sobolev functions with fractional smoothness. An explicit $L^2$-lower bound shows that already lower order quadrature rules, such as the trapezoidal rule and the classical Riemann estimator, are rate optimal, but only the trapezoidal rule is efficient, achieving the minimal asymptotic variance.